Commuting Structure Jacobi Operator for Real Hypersurfaces in Complex Space Forms

U-Hang  Ki; Hiroyuki  Kurihara

doi:10.4236/apm.2013.32038

U-Hang Ki^*, Hiroyuki Kurihara^*

Abstract: Let M be a real hypersurface of a complex space form with almost contact metric structure (φ,ξ,η,g). In this paper, we prove that if the structure Jacobi operator Rξ=(·,ξ) ξ is φ▽ξξ-parallel and Rξ commute with the shape operator, then M is a Hopf hypersurface. Further, if Rξ is φ▽ξξ-parallel and Rξ commute with the Ricci tensor, then M is also a Hopf hypersurface provided that TrRξ is constant.

Keywords: Complex Space Form; Hopf Hypersurface; Structure Jacobi Operator; Shape Operator; Ricci Tensor

Cite this paper: U. Ki and H. Kurihara, "Commuting Structure Jacobi Operator for Real Hypersurfaces in Complex Space Forms," Advances in Pure Mathematics, Vol. 3 No. 2, 2013, pp. 264-276. doi: 10.4236/apm.2013.32038.

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